Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 461: 102



Work Step by Step

Given $$\int_{1}^{4} \frac{d x}{x^{3}+1}$$ Since $\Delta x=\dfrac{b-a}{n}=\dfrac{3}{6}=0.5$, then \begin{align*} T_{n}&=\dfrac{1}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+..+2f(x_{n-1})+f(x_n)\right]\Delta x\\ T_{6}&=\dfrac{1}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+2f(x_{3})+\cdots+f(x_6)\right]\Delta x\\ &=\dfrac{1}{4}\left[f(1)+2f(1.5)+2f(2)+2f(2.5)+2f(3)+2f(3.5)+f(4)\right] \\ &\approx 0.358 \end{align*}
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